► A MEASIBEMENT OP THE Ni^*^ DIE3CTI0NAL COHSELATION WITH A TUMT'RT. DIODE COUTGIDEHGE CIECUIT by GARY EDWIN CT.ARK « B. A., Park College, 1961 A MASTER'S THESIS submitted in partial fulfillment of the * requirements for the degree MASTER OP SCIENCE Department of Physics KANSAS STATE UNIVERSITY Manhattan, Kansas. . 1966 Approved by; Major Professor ■• n C91 TABLE OF CONTENTS I. INTRODUCTION 1 1. Purpose of the Paper 1 2. Review of Literature 2 II. TUNNEL DIODE COINCIDENCE CIRCUIT ........ 3 1. Properties of the Tunnel Diode 3 2. The Uni vibrator Circuit 5 5. Detection of Coincident Pulses 9 4. Zero Crossover 10 5. Noise and Feedthrough ......... 11 6. Slewing 12 7. Effect of Temperature lA- 8. Jitter 14 9. Nanosecond Systems, Inc., Equipment . • 14 III. THEORY OP DIRECTIONAL CORRELATION 1? 1. The Directional Correlation Function . . 17 2. The Ni60 Cascade 20 $. Analysis of Chance Rates 21 4. Experimental Considerations 24 5. Correction for Finite-Size Detectors . . 27 IV. MEASUREMENT OF THE Ni^^ DIRECTIONAL CORRELATION 51 1. Apparatus 51 2. Determination of Resolving Times .... 52 5. Determination of Chance Rates 55 4. Determination of Correction for Finite- Size Detectors 54 5. Reduction of Error at 155° ....... 55 6. Analysis of Ni«^ Correlation Data ... 56 V. CONCLUSION , 41 VI. ACKNOWLEDGMENTS 42 VII. PLATES. . . 45 VIII . REFERENCES 75 I. INTRODUCTION 1. Purpose of the Paper In nuclear spectrometry scintillation materials and photo- multiplier tubes are frequently used which produce electrical pulses with two useful parameters, pulse height and time of occurrence. In this paper we are concerned with the second parameter and in particular a method of determining whether or not two pulses occur coincidently within a few nanoseconds (10"-' seconds) or less. Tunnel diodes, which are sometimes called Esaki diodes, are a suitable choice for a coincidence circuit for several reasons: They have fast switching times, can be very much overdriven without harmful effects, consume little power, and require only a small signal. Furthermore they are insensitive to humidity variations, can operate over a wide temperature range, and withstand neutron irradiation. The fact that they are two-terminal devices, and therefore not unidirectional, is a large disadvantage which can be somewhat overcome by careful circuit design. A txinnel diode coincidence circuit similar to the one 48 described by Whetstone was constructed and its properties are discussed in the second part of the paper. One application of coincidence circuits is in the measurement of a directional correlation, the theory of which is treated in Part III. The experimental determination of the directional correlation func- 60 tion of the Ni cascade using this circuit is given in Paj?t IV. Sufficient evidence is presented to indicate that the circuit can be used for this type of measurement. Before proceeding witb. the main part of this paper a short review of the litera- ture is presented. 2. Review of Literature Scintillation methods have "been employed in nuclear 51 physics since the beginning, while coincidence techniques were used as early as 1929* Although most present day fast coincidence circuits utilize solid state devices, early circuits built by Rossi and others had vacuum tubes only. ^^'^^^ j'^^i'^o, i ^ ^g^ circuits are reported which com- bine tunnel diodes and vacuum tubes or tunnel diodes and 1 55 46 transistors. *^'^' Avalanche transistors have been employed 2 15 but not extensively. * ^ The complexity of coincidence circuits ranges from a simple circuit of crystal diodes to extremely complex combinations of electronic devices. There are many variations of the fairly simple and highly reliable circuit with which the paper is con- cerned. ^^»^^»^^'^5»^'''^ The basic building block (see Plate II) of such a circuit is the tunnel diode univibrator, which can handle coxinting rates in excess of 100 Mc/s pro- 1 49 vided dead time is properly controlled. * "^ Because tunnel diodes have extremely small switching times, this circuit is advantageous when a very small coincidence resolving time is desired. A word must be said about gallixim arsenide tunnel diodes, which were considered very unreliable a few years ago. Gallium arsenide tunnel diodes, when not subjected to continuous volt- age high above the forward point, and germanium tunnel diodes are now equally reliable and are available at reasonable prices. Their various properties will be discussed in the next section. II. TUMEL DIODE COINCIDENCE CIRCUIT 1. Properties of the Tunnel Diode Simply stated, a tunnel diode is a heavily doped p-n semiconductor junction. A lightly doped junction has little resistance when forward biased and high resistance when reverse biased up to the reverse breakdown or Zener voltage. If the concentration of impurity atoms is increased, the reverse breakdown voltage approaches zero; and it might be thought that an infinite concentration of impurity atoms would be needed to attain a breakdown condition at zero applied voltage; however v/hen doping is increased beyond a definite concentra- tion, which is easily obtainable in practice, a reverse break- down condition exists even at a small forward bias, and the device is a tunnel diode. Such a diode exhibits negative resistance at a small forward bias. The static characteristic curve of a tunnel diode is shown in Plate I. The negative resistance region is between the peak point, labeled b, and the valley point, labeled d; and point c is called the forward point. Theoretically this cxirve should be the sum of a Gaussian and an exponential, "but it differs in two ways. First, the current in the valley region is always greater than would be expected, the differ- ence "being due to defect states in the forbidden gap energy o band as postulated by Yajima and Esaki. Second, the peak region is always skewed slightly to the right, this being attributed to stray circuit parameters which can never be completely eliminated. In order to make circuit calculations, the characteristic 29 ^1^1 curve may be approximated by two psirabolas * or a series 22 of straight lines. In the latter approximation the resist- ance in the low voltage region is given by the formula S,, = 0.75 "V /I » where Y^ is the peak voltage and I is the dl PP P P peak cvirrent; and the resistance in the high voltage region is given by R^2 ' ^^fp~ ^v^'^^^n" "^v^ ' where V^ is the forward point voltage, I is the valley current, and ■\r^ « 0.5(V^ - Y^) with "V being the valley voltage. The peak, valley, and for- ward point voltages, which are determined by the semiconductor material, are 55 mV, 359 21V, and 500 mV respectively at 25° C for germanium and about double these values for gallium 21 arsenide tunnel diodes. Measured values of parameters usually agree with those calculated from the linesirized, curve to within ten per cent. The peak current is determined by the impurity concentra- tion and the cross-sectional area of the junction. In prac- tice the junction is made larger than necessary and then etched away until the desired current is achieved. A rather high current density is used to decrease tlie capacitance, which depends on the peak current. A one milliampere unit, for —4- example, might have a junction diameter of less than 5 X 10 inch. The ratio of peak current to junction capacitance, I^/C, is approximately constant for a given semiconductor material and doping, and is about 1/2 for the 1N5118 gallium arsenide and about 5 for the germanium tvmnel diodes utilized in our circuit. Because an input pulse must supply charge to the junction capacitance, and because the rate of charge accumula- tion is a function of overdrive, there is a time delay in the firing of the tunnel diode. The variation in this time delay with overdrive is called the slewing characteristic. The time delay at the input stage is reduced considerably by using a germanium tunnel diode, for which the junction capacitance is about ten times smaller than that of the gallium arsenide tunnel diode. 2. The Univibrator Circuit There are several possible variations of the basic txinnel diode univibrator shown in Plate II: Sometimes the bias volt- age is introduced on the other side of the inductor; occa- sionally there is a resistor between the tunnel diode and ground, and quite often all polarities are reversed. A typical output pulse is shown in Plate III with the letters a, b, c, d, and o corresponding to the same letters in Plate I. The voltage and current under quiescent conditions are given by the coordinates of point o in Plate I. The negative resistance region between b and d is unstable; consequently the tunnel diode switches rapidly to point c to accommodate the input current, thus producing the leading edge of the output pulse. Here precisely, the current may rise above c for large input pulses. The curve is then followed from c to d, thus producing the corresponding portion of the output pulse, where- upon the circuit relaxes to the operating point, o. The wiggles in the tail of the pulse are due to reflections within the circuit and oscillations about the operating point. The maximum voltage (or current) to which a pulse rises is the pulse height. If the initial slope of the pulse is small, and the top of the pulse is fairly flat, it may be difficult to measure exactly when the pulse starts and when it reaches its maximiim. Por these reasons the rise time, t , of a pulse is defined as the time for the pulse to rise from 10% to 90% of the pulse height. If a trigger of minimal ampli- tude is" used with a constant current bias so\irce, for which the load line is horizontal, the rise time of a tunnel diode is approximately given by the relation t » ("V^tj" ^-d^^/^^-d" -^v^ ' where all quantities have been defined previously. The switching time, which is determined mainly by the junction capacitance sind somewhat by the amoxint of charge available from the trigger, of a GaAs tunnel diode with a peak current- of 10 ma and a junction capacitance of 10 pfd is 25 reported to be approximately one nanosecond, ^ and this was verified on a Tektronix Tunnel Diode Rise time Tester. With the tunnel diode connected into a iinivibrator circuit, the rise time of an output pulse is somewhat greater, typically aroxind two nanoseconds or less. In a manner similar to the above, the decay time, t^, is defined as the time for the pulse to decay from 90% to 10% of the pulse height. The pulse width is the full width of the pulse at half maximum, FWHM. Because the circuit is under- damped, the pulse overshoots equilibrium during decay, and the recovery time includes this overshoot. The minimum time between two input pulses which will cause the univibrator to be triggered twice when the second pulse is 10% above threshold is the pulse pair resolution time, t . In the simple tunnel diode univibrator the expression t = 14(FWHM) is valid over a fairly large range. ' Whetstone has shown that by replacing the 13 ohm load resistor with a high conductance diode and resistor combination, the recovery time of the circuit can be controlled so that pulse pairs well under ten nanoseconds apart can be coxinted, but the simpler circuit was considered ' adequate for our purpose. The output pulse height is a function of the inductance in the circuit. Results from four units, which are plotted in IlQ Plate IV, differ from those obtained by Whetstone for several reasons. The pulse height was measured not across the tunnel diode itself but at the point where the pulse entered the coincidence circuit; therefore the pulse heights were smaller than those of Whetstone. The inductance was somewhat difficult 8 to measure due to its small magnitude. The inductance plotted is that of the inductor, as measured with a Tektronix IJO L-C Meter, and does not include the stray inductance. The output pulse height also varies with the input pulse height, and is some constant plus about 5% of the input pulse height. This dependence is greatly reduced hy feeding the output from the first univibrator to a second univibrator to produce a stand- ardized pulse which is almost independent of the input pulse height. Over a range of several volts the variation in pulse heights from a two stage unit was just barely perceptible on a Tektronix sampling scope. There is another advantage in using a two stage unit. For a wide input pulse the first stage may produce a train of pulses, but with careful adjust- ment of the bias on the second stage, it will fire on only the first pulse in the train due to the fact that the first pulse of the train has the greatest pulse height. A complete circuit as shown in Plate 7T has two discrimi- nator units, each with two tunnel diodes, and a coincidence unit with two tunnel diodes. The inductor in the final uni- vibrator has been chosen larger than any of the other induct- ances so that the final output pulse is wide enough to be accepted by succeeding equipment. Through the complete circuit there is a propagation time which measures about eight nano- seconds but depends on the bias settings. The bias on any univibrator stage must be set low enough so that a pulse from the previous stage will trigger it but high enough so that feedback pulses from succeeding stages will have no effect. Of the several input circuits tested, it was difficult to ascertain which one was really best since all were able to fire the univibrator. Because of the threshold nature of the circuit, the univibrator either fires or does not fire, so simply looking at the output does not give a measure of the quality of the input circuit. A transistor input circuit was constructed which ultimately gave good results, but as there were some difficulties in getting the bias voltages for the . transistor initially adjusted, and as the amplification proper- ties of the transistor were not needed, this circuit was regarded as producing unnecessary complications. Pulse trans- formers constructed from ferrite cores according to informa- tion obtained from Whetstone were used in the input circuit and gave acceptable results, but perhaps due to the use of wide input pulses, slightly greater input pulse heights were required. A simple capacitor input circuit was also tested and gave good results with capacitances from 35 uuP to 100 times this value. 3. Detection of Coincident Pulses Consider two standard pulses, identical except for time of occurrence, which are added linearly and fed to a coin- cidence circuit. Because the coincidence circuit is set so as not to trigger on a single pulse height, it will not trigger \inless the pulses overlap and the sum of the overlapping pulses at some time is greater than the single pulse height. 10 This can occur only if the pulses are coincident to within a time of less than the FWHM of a single pulse. In practice the coincidence circuit must be set to trigger at a threshold greater than once hut less than twice a single pulse height. The same principle can also be used to determine if three or more pulses are coincident within a certain time interval. Other methods of time analysis than that of pulse overlap are q sometimes used.^ To measure the resolving time a series of 50 ohm cables of different lengths were made and the time for a pulse to propagate through each was measured with a Tektronix sampling scope. A single source fed the inputs of the coincidence circuit through different cable lengths. By selection of the cables it was determined that the resolving time of the cir- cuit in a typical configuration was between 2.5 and 3 nano- seconds, which was consistent with the standard pulse widths. 4. Zero Crossover Complications arise from the statistical nature of photo- multiplier pulses, the problem being especially acute in the case of slow pulses from Nal(Tl) detectors.^ The question arises as to which part of the photomultiplier pulse gives the best time measiirement. When a quantum of radiation strikes the cathode, a photoelectron is ejected. The electron is accelerated to the first dynode where more electrons are ejected, and so on. Since the electrodes have finite dimen- sions, the paths of different electrons vary slightly in length. 11 and the final anode collects a swarm of electrons which, arrive within some interval of time. The trailing edge of the pulse is highly dependent on the circuit <. j not useful for timing pxirposes. The leading edge of the photomultiplier pulse tells only when the first few electrons arrive at the anode, but we seek the average time of arrival, which is approximately at the time of the maximum height of the pulse. As the top of the pulse may be fairly flat, the time at which it reaches a maximum may be difficult to ascertain. The negative photomultiplier pulse can be dif- ferentiated thus producing a pulse which is initially negative going and then positive going. If the differentiating circuit is properly designed, the zero crossover between the negative and positive going parts of the differentiated pulse will be a good measure of the time of the maximum of the photomulti- plier pulse. The zero crossover signal may also be obtained by the clipping stub method or the LC tank circuit method. Some circuits are so sensitive that they can detect a single photoelectron. In practice our coincidence circuit does not respond at exactly the zero crossover but very soon afterwards. 5. Noise and Feedthrough With a single input signal of 100 mV the feedthrough at the output of the second stage is 0.8 mV, while a single 500 mV signal gave a feedthrough of 1 mV. Noise generated in the cir- cuit was certainly less than 1 mV. Since the output pulse for coincident singles had a height of several hundred millivolts, 12 noise and feedthrough were negligible and did not upset suc- ceeding circuits. An important consideration is the amoxmt of noise and feedthrough. at the coincidence diode. A 100 mV signal at the input produced a signal of 2 mV at the coincidence diode due to noise and feedthrough, and varied almost linearly with increasing input signal. As a smaller input signal produced smaller feedthrough, and as the standard pulse had a fixed height, it appeared best to use the least amount of amplifica- tion needed before applying the signal to the first tunnel diode. The effect of large signals was to increase the resolv- ing time of the circuit. The circuit operated most effec- tively when the input signal was not too much above threshold, say 30% at most. For very severe overdrive the coincidence diode might be expected to fire on a single pulse. With the circuit set to give an output pulse for coincident 100 mV pulses, a single input pulse of 20 volts would not fire the coincidence diode. During one period of counting it was found that many spurious pulses were registering on the scaler at odd intervals. While observing the scaler it was determined that these, pulses always occurred when an electric desk calculator was being used in the same room. Thus the circuit was not adequately shielded from line transients. 6. Slewing The "slewing" or "walking" characteristic is defined as 13 the time variation of the output pulse as a function of excess 46 of the input pulse over threshold. The threshold must be set at a few millivolts above zero, otherwise the circuit will oscillate. For this reason the circuit can never really trig- ger at the zero crossover but will trigger a very short time afterward when the signal has risen to threshold. As the amplitude of the input signal is increased, the slope at the zero crossover increases; because the signal reaches threshold sooner, the tunnel diode would be expected to fire sooner. This is the case for a positive pulse, but for a zero cross- over pulse another factor comes into play. The negative por- tion of the pulse drives the operating point of the tunnel diode down the positive resistance part of the characteristic curve, and it would seemingly take longer for the current to rise to the peak point. Thus there are two competing effects. The greater slope of a higher amplitude input pulse tends to make the tunnel diode fire sooner, while the larger negative portion of the input pulse drives the tunnel diode down from the operating point and tends to make the tunnel diode fire later. The result is a U-shaped slewing char-acteristic as the tunnel diode fires late for both small and large zero crossover input pulses. Plate VIII shows the slewing characteristic as a fxxnction of per cent of threshold. The slewing characteristic of a commercial tunnel diode coincidence circuit is shown in Plates XI and XII. These results sire not quite as good as those which have been reported, even though we used a germanium 14 rather than a gallium arsenide .^el diode at the input stage to improve the slewing charact .stic. 7. Effect Temperature At higher temperatures -.3 effect of intrinsic carrier concentration, the result of thermally created electron-hole pairs, becomes large in comparison to the effect of impurity atoms. In the tunnel diode this results in a changing of both the peak current and the peak voltage. In a germanium tunnel diode the decrease in peak voltage with increasing temperature is about 0.11 mV/C^. In our circuit the tunnel diodes were mounted in a position which allowed adequate ventilation to reduce temperature changes to a negligible amount. 8. Jitter The jitter of the threshold on the input stage is approxi- mately 2 mV for the germaniiim tunnel diode. The minimxim usable threshold without sacrificing stability is approximately 60 mV, but usually the threshold voltage is set at 100 mV or higher. The jitter of the threshold voltage results in a time jitter of the firing of the tunnel diode. This time ji't'ter was determined to be a small fraction of a nanosecond, and ^or the present equipment is small enough to neglect. 9. Nanosecond Systems, Inc., Equipment In the fall of 1964 equipment was p\irchased from Nano- second Systems, Inc. The circuit employed is that of Whetstone^^ 15 with improvements. ' It was found to behave quite similarly to the previously constructed equipment. It possessed many of the same difficulties such as double pulsing on wide input pulses. One good feature was that the commercial equipment was much better shielded and did not respond to line transients from the previously troublesome electric desk calculator. The resolving time of the ITajiosecond circuit was measured in two different ways for various settings of the knob labeled "logic" on the coincidence unit. Table I shows the data obtained by feeding signals from a pulser to two discriminators through different delay lines. The range in time over which one signal could be varied relative to the other was then taken as twice the resolving time of the circuit. As a check the linear sum of the two standard 410 mV pulses entering the coincidence circuit was also viewed on the sampling oscillo- scope. Table II shows the maximum height of this sum. with Table I. Resolving time by the direct delay method. Logic knob Resolving time in setting nanoseconds 3.60 16.3 4.00 14.5 5.50 5.5 7.00 2.0 7.50 .75 various delays in the circuit. The logic knob on the coinci- dence unit is actually a threshold setting knob although it is 16 not directly marked in millivolts. Suppose a particular setting of the logic knob, say 5»00, corresponds to a threshold of 610 mV. Prom Table II we see that one pulse could be varied in relative time from 28 to 48 nsec, which would give a resolv- ing time of 10 nsec* The resolving time at various logic setting was also determined by the chance coincidence method from the data in Table III. The right hand column was calcu- lated from the formula t = N^/(2 N^K2)» where N^ and N^ are the single counting rates and IT is the chance coincidence counting rate. As shown by the plot in Plate IX the results of the two methods are in excellent agreement. Table II. Maximiim height of two overlapping standard pulses. Delay between pulses Maximum height in in nanoseconds millivolts 10 410 16 420 20 510 28 610 30 660 36 810 40 790 48 610 50 550 56 420 60 410 17 Table III. Resolving time by the chance coincidence method. ^ . „ . r, ^- Count rate Coxint rate Resolving Logic Coin. Counting ^^^^ ^^go from pulser time setting counts time ^^^^^^ (^^^^^ {nzec) 3.60 1684 30.00 min 204 21? 4.00 660 30.00 min 204 21? 5.50 312 30.00 min 205 277 7.00 1423 12.00 hr 203 783 7.50 801 17.75 hr 205 ^58 251 682 16. 5 251 592 12.9 251 587 6.1 251 205 1.15 251 789 0.435 III. THEORY OF DIRECTIONAL CORRELATION 1. The Directional Correlation Function Directional correlation measurements involve the coin- cidence counting rate as a function of the angle between the propagation vectors of two cascade radiations. Many treatments of the subject are found in the literature, but we will give only a brief discussion of the theory since this paper is mainly concerned with the experimental problems. For convenience we take the direction of propagation of the first gamma ray as the polar axis of spherical polar coordinates. The angular distribution can be most appropri- ately described by a series of spherical harmonics: WC0,0) -^aj^y^(O,0), (1) L,m where WC©,0) is proportional to the coincidence counting rate as a function of the direction of propagation of the second gamma ray. For randomly oriented nuclei and the special polar axis 18 we observe that the distribution must be independent of the azimuth angle. In this case m=0, and the series may be simplified: W(0,0) -Ea^-Q I°(9,0) - W(©) W(0) = EAl P^Ccos G) (2) L where G is the angle between the propagation vectors. A gen- eral derivation with an arbitrary orientation of the coordin- ate system gives the same result. Two more simplifications can be made. If the transitions involve states of definite parity, only the even Legendre poljmomials appear. Since the transitions are between eigenstates of angular momentum, the maximum value of L is limited by certain triangular inequalities: L/2 = Min(L^, I, Lg) (5) where L, and Lp aire the angular momenta of the radiations and I is the angular momentum of the intermediate state. Of course this series could be expressed in terms of another set of functions such as powers of cos G, but a par- ticular advantage of Legendre polynomials is that each coeffi- cient can be split into two factors, each of which depends on only one of the transitions. For pure transitions, ^L " ^L^^l ^1 ^i ^^ ^L<^^2 ^2 ^f ^> ^""^ where I. and I-. are the angular momenta of the initial and final states. These factors are certain combinations of r 19 Clebsch-GordorL and Racaki coefficients and have been tabulated in various references. Note that V(0) is symmetric with respect to an interchange of the order of the transitions. If a transition is mixed with a mixing ratio, d, defined by ,2 intensity of (L + 1) ^5^ ^ ' intensity of (L) ' ^^^ then the factors become ——2 Ve now derive these functions for gamma-gamma angular correlation in a more explicit but less useful form. Ini- tially a nucleus may be in any of the- 21^^+ 1 equally populated initial substates with spin I^ and projection quantum number M- . The first gamma ray is characterized by quantxim numbers L, and M, . For the special choice of polar axis M^ can be only +1 or -1. The intermediate state is characterized by I and M. These three sets of quantum numbers must satisfy the vector addition Xi^ . IL^ -*• n-MT (7) f From the definition of Clebsch-Gordon coefficients Thus the relative transition probabilities are given by u 20 and the relative populations of the intermediate substates are Ij (IL^^m^ ^i^i)^ (10) "l where we have summed incoherently since the phases are random, and the prime indicates that M, is only +1 or -1. The angular distribution of the first gamma ray is then A__j (IL^m, l.n.)\'^iQ) (11) where P^ (9) is some angular function characteristic of the ■^1 radiation. Por a radiating multipole of order L and projec- tion M p'^CO) = &(L-M)-H(M-1)3 |y/-^|2. 2H^1y^|^.[l(L^1)-H(M.1)]|y^^^|^ L 2 L CL + i; (12) Since the first gamma ray is observed only at 0=0, the corre- lation function is , , (IL^MM^ I^M^)^P^^(0)(I^L2M^M2 IM)^F^2(0) I^,M,M^ 1 -2 M, M^= +1 or -1 V ^^f -M , (13) 2. The Ni^^ Cascade The standard nucleus for directional correlation measure- ment is Ni . The nucleus Co decay to Ni by negatron emission with end point energies of 0.513 Mev and 1.45 Mev with very nearly 100?^ of the transitions having the former end 21 60 point energy. These transitions lead to a Ni nucleus in a 2.5 Hev excited state as shown in Plate XIII. The groiind state is reached "by emission of two cascade gamma rays of energies 1.1? Mev and 1.35 Mev. The 1.55 Mev gamma ray has been established as the second transition by means of nuclear flourescence resonance experiments. The spin sequence is well established as 4(E2)2(E2)0 with both transitions being pure electric quadrupole. All three levels have positive parity. 60 The theoretical directional correlation function for the Ni cascade assuming the above spin sequence is W(0) = 1+0.102 PgCcos 0) + 0.0091 P^Ccos 9). (14) 5. Analysis of Chance Rates Before comparing experimental data with the theory pre- viously described, several corrections must be made. We must first subtract chance coincidences between unrelated pulses. The analysis of chance coincidences is treated extensively in the literature.27.5'^.59.« In a double coincidence circuit the chance coincidence rate is easily shown to be N^ = 2 t(N^- lft^^^2"^t^ ^"^^^ where t is the resolving time, N^ is the true coincidence rate, and N^ and Np are the singles counting rates in the two chan- nels. In practice the true coincidence rate is always so very much smaller than the singles counting rates that the N^ terms may be neglected. In a triple coincidence circuit of the fast-slow type 22 there are several possibilities for chance counts. The various singles and coincidence rates are shoi«i in Figure 1. It is assumed that the singles rates, Kj and ITj^, from the fast discriminators are larger than the corresponding singles rates, N, and Np, from the single channel analyzers. PM 1 PM 2 disc 1 -1 disc 2 ^1 N II Ni SCAl JJ« SCA 2 fast coinc slow coinc t3 N Fig. 1. Block diagram of a fast-slow coincidence circuit. For simplicity it is assumed that the coincidence circuits operate in the following manner. A single pulse input to the fast circuit produces a rectangular pulse of width t^. If two pulses from the two fast inputs overlap at any time, an output pulse is produced. The slow circuit works in a similar manner except that pulses from all three inputs must overlap at some time in order to produce an output pulse. The chance coincidence rate "between N, and 1^2 o^l7 is given by -1 25 1T(1 S: 2) = (t^+ t2)M2 (16) with, an average overlap time of t,t2/(t^+ tg)* The triple chance coincidence rate is then = (t^t2+ t^tj+ t2t^)lT^ir2N5 (17) Por t,= tp" t^=» t, the preceding expression is N^(lSi2Si5) - 3 t^ ^^^211, (18) The fast coincidence rate, N"^, can be split into several parts; where N. is the true coincidence rate between gamma rays of the proper energies, N^., is the fast coincidence rate where only the first gamma rays have the proper energy, N^2 1^ "''^^ rate with only the second gamma rays having the proper energy, N» is the fast coincidence rate between gamma rays, neither of which has the proper energy, and the la^t term in (19) is the chance coincidence rate. The triple slow coincidence rate, N, can be divided into eight parts: N - 11^12^ Nfi(2)-^ ^f(l)2^ ^f(l)(2)* ^(f)12^ ^(f)l(2) *^(f)(l)2-' ^(f)(l)(2) (20) where parentheses indicate a component which is in slow coin- cidence but not fast coincidence with the other components. As an example l^fn /•p') ^^ *^® coincidence rate between N^, and a gamma ray of energy E2, the latter being in slow but not r- 24 fast coincidence with the former. The nature of the various terms is easily deduced from the following formulas: "^1(2) - 2 t N^^Nj (21) "f (1)2 - 2 * ^tzH (22) ''f(l)(2) - 5 *^ V2''f (23) . ''Cf)l(2) • ^2 Vii%) 2 t Hg (24) K(f)(l)2 - ^2 tAH2) 2 t S^ (25) K(i:)(l)(2) - 5 *^ V2 2VAI (26) Equation (20) now becomes N - 1T^+ 2 tCNf;L^2"' ^f2^1^ + 5 t^ N^IT2^^+ 2 t^3_lT2 + 4 t^t N]_N2^^I+ %I^ + 6 t^ *f^lN2^I^II- (27) 4. Experimental Considerations The singles counting rate in either detector is N. . Mp.e.n. (28) where M is the source strength in disintegrations per unit time, p. is the probability that a disintegration will lead to the gamma ray of interest, e^^ is the efficiency of the detector, and n. is the solid angle subtended by the detector in units of 4 ff. The accidental or chance counting rate in the coin- cidence circuit is No" 2 t e^NilT2 - 2 t e^Mp^e^n-LMp2e2n2 (29) where e is the efficiency of the coincidence circuit, c The 25 true coincidence rate is N^= M PiP2eie2e^nin2 K(9) (50) where K(0) is proportional to the directional correlation function and has an average value of unity. The true to chance ratio at any angle is ^ ^ M PlP2^c^l^2^1^2^^Q^ (51) ^c ^ ^ ®c^ Pi^i^l^ P2e2^2 so that the average value is CWave - ^/^2 t^). (32) The actual counting rate is the sum of the true and chance rates from which the chance rate must he subtracted. In order for the chance rate to contribute only a small error to the results the true to chance rate should be, as high as possible. In practice there is a lower limit for the resolving time of the coincidence circuit due to electronics. Furthermore if H is small, the counting rate will also be small, but it is desirable to have the singles rates many orders of magnitude above the background rate. A good compromise is to have the true to chance rate between five and twenty. As a numerical example, let N^/N^» 5 and t^ - 25 nsec. Then from (52) M - lT^/(2 t^N^) « 0.1 mc (35) Also, let e^^- e^- 0.05, n^=^- 0-01, and e^- 1. From (29) and (50) N a 4.5/min c N^= 21.5/min (5^) 26 Suppose we wish to know the counting rate to within the frac- tional statistical error E, and there is no error in the tine T. e2 »t*c* %^/''K ^> For E » 1%, T » 600 min at each angle. Por E » 9.1%, then T = 60 000 min at esich angle. As will be shown in the next section, for coimters which subtend a finite solid angle, the directional correlation function is attenuated. For this example U(9) = 1 + 0.100 P^Ccos ©) + 0.0086 P. (cos 9) (36) with a few values being WC90^) = 0.9532 W(135^) - 1.0215 W(180^) » 1.1086 (37) In order to propagate the errors we need White's matrix: 42 56 7^ I^O0°)\ /Aq\ I05 I -90 40 50 48 -96 48 7 / N(155°) 1 N(180°) / \^4/ (38) Suppose we wish to determine A^ to within ten per cent,, then from (58) the counting rate of true coincidences at each angle must be about 0.15% correct. From (35) the counting time at each angle is twenty days I If both counters accept both gamma rays, only half this time is required. We see that a tremend- ous amount of time is required to collect directional 27 correlation data. 5- Correction for Finite-Size Detectors When the directional correlation is observed experi- mentally, it appears "smeared out" according to W(0) = — (39) /y dn^dn2e^e2 The efficiency, e, of a detector has been treated in several 17 ways. Feingold and Frankel ' use an expansion in spherical hairmonics for the efficiency of an arbitrary detector e(0,0) = C^^^''^Lml^(G,0) (^) L,m For a circularly symmetric detector as shovm in Figure 2, this expression simplifies since e is independent of 0. eCG) = Ea^ P^-Ccos Q) (41) The expression which Rose^' uses for the efficiency is e » 1 - exp(-kx), where k is the absorption coefficient for the gamma ray energy of interest and x is the detector thickness at the angle of incidence of the gamma ray. For a circularly sym- metric detector X » t sec b for 0^ b ^tan"-'- [r/(h+t)] - b' X = r CSC b - h sec b for b' ^ b ^ tan" (r/h) (42) Since the directional correlation function is most often expressed as a series of Legendre polynomials, we must perform 28 detector 1 detector 2 Figure 2, Angles involved in correction for finite- size detectors* • 29 integrations of the form I ^ J^y dS\^dS\^e^e^jJ^QQs ©')• (43) Ve express P^Ccos 0') in terms of b^ and 0" "by means of the addition theorem for spherical harmonics. +L P^Ccos e-) = 5^C YS'(t,,0) YS(e",0") m=-L (44) Let X, "be independent of 0^ so that only terms with m»0 con- tribute to the integral. Then P^Ccos 0') - ^ YO*(bi,0)Y2(Q",0") + ... » P^(cos 0') - Pj^Ccos b^) P^Ccos Q") + ..., (45) and the integral I becomes / d0^y^sin bj_db^Pj^(cos 'b^)e^/j'^0^3±a, Q' "d0"Pj^(cos 0")e2 (46) The efficiency of the second detector is most < easily expressed in terms of angles measured from the Zp axis. Ve apply a rotation matrix and drop the m^O terms which do not contribute. This gives i P^Ccos ©") - Dq5(0,©,0) P^Ccos b2) + ... » Pj^Ccos 0) Pj^Ccos bg) + ... i The integral I becomes I - 4ii^ Zjil) Jl(2) Pj^Ccos 0), ' 30 where - Ji,(i) 1 « / sin b,db^P-j^(cos l^i)©]^ Jl(2) ( = / sin b2db2PT (cos b2)e2« (47) The experimentally observed directional correlation function then becomes • W(0) = 1 + (^22^2^2^^°^ ®^ "^ Q44^4?4 (cos 0) + ., • • > where f Q22 = . Q2CI) ^2^2^ Q2CI) 1 = J2(l) / JqCD. (48) As an approximation we assume that e is constant up to b^ and zero thereafter. h- ♦ ^b / Pt-(cos b)e sin b db Jo- ey sin b db « e(l - cos b ) ^2- e /P2(cos b) sin b db » e cos b^ (1 - cos2b^)/2 J4" e /p^(cos b) sin b db - 2 2 e cos b (1 - cos b )(7 cos b - 3)/8 (49) Then the attenuation factors for similar detectors < ire Q22 = ' (Q2)^ 1 ^2- COS b^ (1 + cos b )/2 «W " ■ (Q^)^ «4" cos b^ (1 + cos b )(7 cos b - 5)/8 (50) \- tan~-^(r/h). These are the formulas given "by Frauenfelder and Steffen 52 As pointed out by Lawson and Frauenfelder, the 31 18 attenuated directional correlation fxinction is of the form derived above only for centered point sources and circularly symmetric detectors. Otherwise a mixing of the different At terms occurs. IV. MEASDEEMENT OF THE Ni^^ DIRECTIONAL CORRELATIOIT 1. Apparatus A source, roughly of strength 0.1 mc as will be shown later, was prepared by evaporating a Co Clp solution inside a carbon cylinder of outside diameter 5/16 inch and inside diameter 3/52 inch. The wall of the carbon cylinder consti- tuted an absorber of 0.575 gm/cm , which was more than suffi- cient to absorb all the beta particles from the 0.515 MeV beta decay, for which only 0.086 gm/cm was needed. The total absorbing material consisting of the cylinder wall and the cover on the Nal crystal was not quite enough to absorb completely all beta particles from the 1.^ MeV beta decay, 2 which would have required 0.7 gm/cm , but no additional absorber was introduced since this decay is only 0.01% as intense as the 0.515 MeV decay. The fixed detector was a ITal(Tl) scintillation crystal 2 inches by 5 cm thick placed 9»2 cm from the source, and the movable detector was a Nal(Tl) crystal 1 5/^ inches by 5 cja thick placed 8.4 cm from the source. Availability was the 32 only reason for choosing different sized crystals. Each crystal was mounted on an EGA 65^2 photomultiplier tube operated at 1100 volts and placed on a specially constructed circular table with the source at the center. Each crystal was surrounded by a very thick conical lead shield. Fast coincidences between pulses from the photomulti- pliers were detected in the Nanosecond Systems equipment and fed to a triple slow coincidence iinit built by David Draegert. Photomultiplier pulses were also directed through a Nuclear Data 500 dual single channel analyzer to the slow coincidence unit. The fast-slow coincidences were counted on a scaler. Simultaneously singles counts passing through the Nuclear Data 500 analyzers were counted on scalers. 2. Determination of Resolving Times A 1.5 uH inductor was placed in the fast coincidence logic circuit. The resolving time of the fast circuit was deter- mined by the chance coincidence method. Two rather long inins were taken giving the following: Nj » 64 927 808 / 856 min N-j-j = 18 187 418 / 856 min N » 1205 / 856 min. o Using (15) the resolving time of the fast circuit is *f " \/^^ Vll^ " ^^'^ i ''^ ^®®*^* The logic setting used was 5 •70. Note that the graph in Plate IX cannot be used here because the inductors were not of 55 the same value, but from Plate H and the graph of Plate V, we can conclude that the result is entirely consistent with previous measurements. This resolving time will be used to calculate expected chance rates. The resolving time of the slow coincidence unit was determined by the same method. Nj_ = 505 5^ / 5 mill N^ = 9^ 612 / 5 min N = 507 / 5 min c The resolving time of the slow unit was then t = 1.60 + .09 usee. This was consistent with oscilloscope observations of the waveforms in the circuit which showed the, pulses fed to the slow coincidence unit to be about 1.5 usee wide. 5, Determination of Chance Rates The measured true plus chance slow coincidence rate between K, and Ng was about 105 cpm of which 91 cpm was the expected chance coincidence rate and 74+8 was the measured chance rate. The measured true plus chance fast coincidence rate between Nj and ^-^j was about 55 cpm of which 5*7 cpm was the expected chance rate and 5.8 ± 'S cpm was the measured chance rate. Prom this a very rough calculation can be made for the source strength from (52), which gives M - 0.07 mc. At this stage, however, the data are much too rough to give more than an order of magnitude. A more precise measurement 54 showed the source strength to be 0.13 mc. Measurements were taken of the various counting rates and approximate calculations were made in order to obtain some idea of the relative contributionn to the chance counting rate, The data used were: N^ - 20 000 cpm N, » 8.? cpm ^2 " 100 000 cpm isr^^ - 1.9 cpm % = 35 000 cpm lf^2 -10.7 cpm Njj = 120 000 cpm t = 1 usee t^ - 25 nsec. Calculation of the terms in (27) gives: 2 t(lT^^lT2+ ^£2^^) - 0.007 cpm ^ 3 t^lT^N2N^ = 0.00000016 cpm 4 t^t N3_N2(N-|.+ ITj-j.) - 0.009 cpm 6 t^t^N^N2lTjNjj = 0.00000016 cpm 2 t^N^N^ =1.7 cpm. It was apparent that the last term made the only significant contribution, so all other terms were neglected. 4. Determination of Correction for Finite-Size Detectors 22 A Na source was mounted on the coincidence table and coincidence rates were taken at nine points about 180°. The annihilation radiation consists of coincident 0.5 MeV gamma 35 rays at an angle of 180°. The chance coincidence rates were subtracted, and the data were plotted. A smooth curve was drawn through the points. The half width at half maximum was measured to be b « 5° 15' + 5»5'' Each detector was then assumed to subtend a cone of solid angle 0.027 steradian. Note that the energy of the annihilation radiation was 60 not the same as that of either gamma ray from Ni . A better but more complicated procedure for finding the half-angle b experimentally would have been to use a well collimated beam of the same energy as each of the gamma rays. Solid angle correction factors were calculated from (50) . Q2Q2= 0.9823 + .0005. (^Q^= 0.9^20 + .0001. 60 The theoretical directional correlation function for the Ni cascade (14) is attenuated to W(0) - 1 + 0.100 P2(cos 9) + 0.0086 P^(cos 0) 5. Reduction of Error at 135° Since there are no odd Legendre polynomial terms in the directional correlation function, it is symmetric about 90° and 180 . This means that the slope is zero at these two angles thus making them ideal experimentally since a small ' systematic error in the angles contributes very little error to the measurement. At 135° > the other standard angle, the slope is not necessarily zero. Indeed in the case of Ni this is quite neair the point of inflection where the slope 36 is maximum, so that a small error in the angle setting at 135° may contribute greatly to the error ir, the measurement. It was thought that a systematic error at 135° could be eliminated by taking part of the me-' ^urements at 135° and part at 225 and adding the counts as if they were all taken at 135 • It was found that the counting rates at these two angles differed by less than the statistical variation after a total of almost 20 000 counts had been taken; nevertheless these measurements were still considered to provide an addi- tional check on the operation of the experimental apparatus. There are, of course, two other angles that might be used in this method, namely 45° and 315*^ » but these cannot be realized experimentally at the present time because of the apparatus . 60 6. Analysis of ITi Correlation Data 60 The Co source was centered on the coincidence table so that the counting rate in the movable detector varied not more than one per cent over the range of angles used. A singles spectrum was observed from each detector. The discriminators on the fast unit were set just above the Compton edge of the 1.17 MeV peak. The single channel analyzers of the slow unit were similarly set. In this way both photopeaks in both channels were used, and the true coincidence rate was effec- tively doubled. After the initial setup procedure a series of runs with equal to 90°, 135°, 180°, 225°, and 270° were taken in which 57 Table IV. Directional correlation data. Angle Time Co\)nts from Counts 5 from True plus between in fixed movable chance detectors min detector detector counts 90° 15.88 269 524 900 005 84 135^ 15.69 264 609 900 000 71 180^ 15.97 270 537 900 001 82 225° 14.25 288 582 900 000 74 270° 14.08 288 Oil 900 001 79 225° 14.28 288 597 900 001 72 180° 15.92 281 750 900 000 65 180° 14.05 284 747 900 000 77 155° 15.7^ 276 491 900 001 74 90° 15.94 278 090 900 004 62 90° 15.90 276 871 900 002 73 155° 15.74 271 586 900 000 81 180° 15.90 275 289 900 001 87 225° 14.18 281 495 900 000 75 270° 15.82 289 448 900 001 65 225° 14.21 295 566 900 001 65 180° 15.92 289 850 900 002 74 155° 15.87 288 255 900 002 68 90° 14.24 288 115 ' 900 002 62 155° 14.25 288 446 900 000 84 180° 14.72 298 850 900 001 90 225° 15. 06 506 215 900 001 67 270° 14.71 512 021 900 002 75 225° 14.92 314 948 900 001 79 180° 14.56 308 442 900 001 80 155° 14.28 300 405 900 001 92 90° 14.55 297 049 900 001 83 155° 14.17 290 466 900 002 95 180° 14.54 297 857 900 001 86 225° 14.86 305 828 900 002 89 180° 14.44 296 997 900 002 97 155° 14.17 290 520 900 005 97 90° 14.14 290 789 900 001 84 135° 14.11 289 007 900 005 75 180° 14.42 296 548 900 001 77 225° 14.73 503 552 900 002 85 • 58 a total of 67 752 coincidence counts were obtained in 8275.23 minutes of actual counting time. Intermittently between these runs various checks were made. Singles rates were counted for short intervals, waveforms ■• .-e checked on an oscilloscope, and spectra were taken on a multichannel analyzer. Table IV shows a portion of the data. In order to reduce the effect of a slightly non-centered source, it is standard practice to normalize the coincidence rates by dividing by the appropriate singles counting rates in the movable counter for each angle. It was thought that if data were taken up to a preset number of counts in the movable detector, this normali- zation would be unnecessary for a quick check on the data. Note that the scaler had some difficulty in turning off at exactly 900 000 counts. If the counting rate in the movable detector is calculated for a particular angle, it is seen that in general the counting rate decreased with time. These data were all taken within a ten hour period, which. is extremely short compared to 5-3 years, the half-life of Co^°. The change in counting rate was apparently an electronics instability. 60 Table V. Ui directional correlation data. angle time 0^ counts 0° counts *^® + chance ,?^o-270o 15^S.40 min 30 676 393 134 8^3 117 11 684 }ilo-^^^ S§§^-J^ ^^^ 50 870 273 217 082 741 19 835 180 2725.66 min 53 951 282 239 154 205 23 231 39 The wide variation in the number of counts in the fixed detector is expected since the statistical error is composed of two parts. The fixed detector counting rate varies statis- tically in itself, and the time varies statistically due to taking a preset number of counts in the movable detector which has a statistically varying counting rate. The analysis went as follows: From the singles rates, the times, and the previous determination of the resolving time, the number of chance covmts was calculated for each angle and subtracted from the right hand column of Table V giving the numbers of true coincidence counts for each angle. The background rate in the movable detector, taken before the source was placed on the apparatus, was 665 cpm. The total number of expected background counts was calculated and subtracted from the number of counts from the movable detector for each angle. The resulting numbers were divided into the respective numbers of true counts to normalize for a slightly off center source. Errors were propagated according to the standard methods. Three numbers were obtained which were proportional to the normalized true coincidence counting rates: NOO'') = (709 ± 9) X 10"^ « N^ 2 1^(135°) = (756 + 7) X lO"*^ - N 17(180°) - (817 + 7) X 10""^ - N, These numbers are proportional to the directional correlation function. A best constant of proportionality was determined 40 from a least squares fit to W(©) at the three angles. - -||- =C2(V k W(©^) )(-W(©i) ) Then we obtain k » 0.0758. The experimentally observed values • of the directional correlation function at the three angles are: W'(90°) » 0.9634 + .0117 W'(155°) ' 1.0204+ .0099 W'(180°) = 1.0988 + .0094 These numbers were fitted by using White's matrix (38) to yield the experimentally observed directional correlation 60 • function for the Hi cascade as W'(0) = 1.00 + (0.094 0.011)P2(cos 9) + (0.0087 0.010)P^(cos 0). This is to be compared with the expected correlation function derived in section IV. 4, W(0) = 1 + 0.100 P2(cos ©) + 0.0086 P^(cos 0). Although the errors are rather large, the directional correlation fxinction can provide very useful information. Por instance, it can be shown that if the first gamma ray is mixed, the error in Ap limits the intensity of any M3 component to less than 0.03%. 41 V. CONCLUSION A tunnel diode fast coincidence circuit was constructed and the various properties determined to be as reported in the literature. A commercial circuit of a similar design was used in a directional correlation experiment, and the results were shown to agree with theory. It is concluded that with proper attention to the inherent problems, such as double pulsing, the equipment can be used in this type of experiment. 42 VI . ACKNOVrLEDGMENTS The author takes this opportunity to acknowledge and thank Dr. Louis Ellsworth for his guidance and assistance in completing this work. The author also wishes to thank Dr. Charles E. Mandeville and Dr. V. R. Potnis. 44 CvJ :n > > ^ > > > a > + CM ' 1 Q. iD D a N > > • O EXPLAlTAa?ION OF PLATE II The basic tunnel diode iini vibrator. -10 V 46 PLATE II 500 ohm vVW 13 ohm 430 ohm -aAA/V GaAs r tunnel diode GE 1 N 3118 germanium diode ID3-050 EXPLANATION OF PLATE III Output pulse of a univibrator. if8 , i ; ■3 i O = ( • —J 0. ( T3 OJ time ^<C , 1 { ■» ~~~~- JD ] o i O •] S;OA EXPLANATIOIT OF PLATE IV A plot of pulse height vs. inductance for the 'univihrator. EXPLANATION OF PLATE V j A plot of the base width of the I output pulse of a univibrator versus the inductance of the inductor shown in the circuit. 52 ,■ > \ V O \ \ 9 - .60 .80 1.00 1.20 1.40 inductance L (jjH) > \ / .40 u < \ . .20 1 1 1 1 1 \ 1 ■^ Cv) 9 00 to T CJ (D9S U) g mpiAA aseq ■ ' ■ J EXPLANATION OP PLATE VI The complete coincidence unit using six tunnel diode uni vibrators. E3a?LANAa}I0N OF PLATE VII (upper) Nanosecond Systems fast coincidence modules, (lower) Fast coincidence unit constructed in the laboratory. 56 EXPLANAOJIOIT OP PLATE VIII A plot showing the slewing characteristic of the coincidence \init constructed in the laboratory. 58 ■■ 3 . 5 > UJ < -J a. \ \p unit B threshold ^SCX) mv 1 r . 190 220 • « . \ 100 130 160 per cent threshold 1 1 1 t > o o CO (DOS U) o o c EXPLANATION OF PLATE IX A plot of the resolving time of the Nanosecond coincidence unit versus the setting of the logic knob. The inductor used was 1.00 uH. \ The resolving time 'was determined \ by two methods and it is seen that I the results of the two are in good agreement. 60 X u < -J CL direct delay 1 chance nnethod I 6.50 7.50 • < / 5.50 setting / logic / 4.50 </5 O ^ ID CO 1 1 1 1 1 1 1 1 1 1 • OcQ^"^C\JOoOCO\rCM (D9S U) 9UU|:^ 6U!A|OSGJ EXPLANATION OF PLATE X A plot showing the output pulse height of the pulse'r when fed into a 50 ohm line. EXPLAITATION OF PLATE XI A plot showing the slewing characteristic of the Nanosecond equipment including amplifier for various thresholds. The abscissa is the input pulse height before amplification. The curves are labeled with the setting of the discriminator knob* 6k X u < _1 a. u (J) (D I 3, CM 00 CO L. o o o C X h 3. £_ o U in 4 o o Cvi O 10 > E o en Q. O o o o o o o f^ to 10 -^ CO CM (Dss u) v;iL|s auui; EXPLANATION OF PLATE XII The same data used in Plate XI are replotted with 'the ahscissa as per cent of threshold. 66 X U I CM D £_ 00 o vO (J -f-" • \— o c o 00 ^— •* en E I LlI u h- <J) o u ■o T-= < (r < X u o z: < LU Q. Ul O 6 >/ / > 1^ t ^ y 1 o in / o K/ o I o to L O 10 1 o L O CO -JL O CM O o o CO o in o o CM O o - o EXPLANATION OP PL/iTE XIII 60 The decay scheme of Co showing the two gamma rays of interest. All energies are in Mev. Although the spins of some states of Ni^O are still iincertain, the energies, spins, and parities of the states involved in the directional correlation measure- ment are well established. PLATE Xlll 68 ,(+) v3. /2. S'^ w Co J 27 0.313 -/00%. i.'^a 0.0/%. H i fn-^) \^ / £+ \ U7d A , f *-~ 1.332 0^- > ' 2.623- Z.286 /.33 2 0<.i/e ,^N.-|^ EXPLANATION OF PLATE XIV 60 A singles spectriim of Co showing the two photopeaks. The arrow shows the approximate setting of the discriminators. The extra peak to the right is a sxim peak due to the high counting rate used. 70 > X u 1— < Q_ i \ • CHAMNEL ^ N- ) o u o -a 1 1 ' O — O 2 i n 1 1 giHOOO ] "ii EXPLANATION OF PLATE XV A plot of the directional correlation data. 73 VIII. HEPEEENCES 1. Adler, A., M. Palmai, and V. Perez-Hendez 100 Mc/s tunnel diode discriminator and pulse shaper. Nuc. Instr. Meth. 15: 197. 1961. 2. Artiges, H. C, and J. C. Brun Circuits a coincidences rapides utilisant des transistors a avalanche. J. Phys. Had. 22:53. 1961. 3. Bay, A. A new type of high-speed coincidence circuit. Eev. Sci. Instr. 22:397-^0. 1951. 4. Bay, A. , and G. Papp « _q Coincidence device of 10" -lO"-^ second resolving power. Rev. Sci. Instr. 19:565' 19^. 5. Beller, L. S. N-sec scintillation coincidence spectrometer system with high reliability. Hev. Sci. Instr. 34:1001-1006. 1963. 6. Biedenharn, L. C. , and M. E. Rose Theory of angular correlation of nuclear radiations. Phys. Rev. 25:729-777. 1953- 7. 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Prauenfelder The correction for finite angular resolution in directional correlation measurements. Phys. Rev. 91:649-652. 1953. 33. Lundby, A. Delayed coincidence circuit for scintillation counters. Rev. Sci. Instr. 22:324. 1951. 34. Mayer-Kuckuk, T. , and R. Nierhaus Uber die bestimmung der zufalligen koinzidenzen in schnell-langsam koinzidenz-anordnungen. Nuc. Instr. Meth. 8:76-78. I960. 76 35 • McGervey, John D. A pulse resolver using tunnel diodes. Hue. Instr. Meth. 14:551-352. 1961. 36. Miller, Barry Tunnel diode applications investigated. Aviation Week 71:72. 1959. 37. Rose, M. E. The analysis of angular correlation and angular • i distribution data. Phys. Rev. 91:610-615- 1953. j i 38. Schram, E. , and R. Lombaert ] Organic scintillation detectors. London: Elseview, I 1963. I 39* Shera, E. Brooks, K. J. Casper, and B. L. Robinson ] Analysis of chance coincidences in fast-slow coin- ^ cidence systems. Nuc. Instr. Meth. 24:482-492. 1963. i 40. Shera, E. Brooks Further comments on accidental coincidences in \ fast-slow coincidence systems. Uuc. Instr. Meth. ] 12:198. 1961. 41. Smaller, B. , and E. Avery • ) The use of gated-beam tubes in coincidence circuits. Rev. Sci. Instr. 22:341. 1951. ' 42. Sprokel, G. J. A liquid scintillation counter using anticoincidence shielding. IBM Jour. Res. Dev. 7(2): 135-145. 1963. 43. Strauss, M. G. Timing slow pulses for fast coincidence measurements. Rev. Sci. Instr. 34:1248. 1963- 44. Tarnay, K. The maximxim power output of the tunnel diode oscillator. Proc. IRE 50(10) : 2120-2121. 1962. 45. Van Zurk, R. Circuit discriminateur d' amplitude utilisant diodes tunnel. Nuc. Instr. Meth. 16:157-162. 1962. 46. Ward, C. B. , and C. M. York A nanosecond pulse height discriminator. Nuc. Instr. Meth. 23:213-217. 1963. 77 4-7. Whetstone, Albert L. Improving the tunnel diode univibrator. Rev. Sci. Instr. 54:412-4-13. 1963- 4-8. Whetstone, A., and S. Kounosu Nanosecond coincidence circuit using tunnel diodes. Rev. Sci. Instr. 33:4-23-4-28. 1962. 4-9. Yonda, A. H. , R. Sugarman, and W. A. Higinbotham 100 Mc counting system. Nuclear Electronics Vol. Ill, Vienna: International Atomic Energy Agency, 1962. pp. 3-13. A IffiASUREMENT OF THE Ni^*^ DIEECTIOITAL COREELATIOH WITH A TUMEL DIODE COINGIDEUCE. CIRCUIT , by GARY EDWIN GLAEK B. A., Park College, 1961 AN ABSTRACT OF A MASTER'S THESIS submitted in partial fulfillmeat of the requirements for the degree MASTER OF SCIENCE Department of Physics KANSAS STATE UNIVERSITY Manhattan, Kansas. 1966 A coincidence circuit with a resolving time of a few nanoseconds has been built using tunnel diodes. The theory of operation is discussed. The circuit consists of a series of tunnel diode uni- vibrators. Germanium tunnel diodes are used at the inputs to give greater sensitivity and to reduce the time jitter. The first stage of each input is followed by a gallium arsenide tunnel diode univibrator. The purpose of this second stage is to provide a standard pulse of typically 5 usee width and 100 mV height which is independent of the input pulse height. The standardized pulses are then fed to a coincidence univibrator through a germanium fast diode which serves to make the circuit unilateral. Standardized pulses from two separate circuits are added linearly in the coincidence unit. The coincidence t\innel diode univibrator is set to fire when the standardized pulses add to give a sum pulse of a certain height, which must be between one and two times the height of a singles standard pulse. The coincidence unit is followed by another xinivibrator which produces a much wider pulse. The output can then be counted by a scaler or used to gate a multichannel analyzer. A fast germanium diode is used to make the circuit uni- directional. With germanixim tunnel diodes in the circuit the voltage changes would be too small to effectively forward bias the fast diode, and the coupling would be primarily through the capacitance of the fast diode, which is bidirectional. For this reason gallium arsenide tunnel diodes, which, have a higher voltage change, are used. The directional correlation function of the well known Ni transitions was determined experimentally using the txinnel diode circuit. Corrections for detectors of finite size were made. The results agreed well with theory. Various experimental checks were made of coincidence rates, resolving times, and waveforms. All experimental results were found to agree well with theoretical predictions.